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November 02, 2012

De-mystifying Mathematics (Part 2) – India’s contribution of zero to the number system

We discussed in the earlier post that mathematics can be
thought as a collection of many logical/ numerical concepts and processes that
make the base for applied sciences and arts. We will touch upon the various
concepts in this series – may not be in the linear flow of typical text books.

One of the first uses of mathematics that we learn and
encounter is numbers – for counting and basic arithmetic. We usually start
associating a written representation with a number as soon as we start learning
them. In my opinion, this association itself causes limitations in later stages
of learning and restricts innovation and creativity!

Let us look into this in more detail. If I say the number
‘Twelve’ – we immediately think of the representation ‘12’. What if I had a
symbol for twelve – say ‘µ’? I could continue to create symbols for each number. We have all played code games in our
childhood. Real counting is also like that – a bit more evolved and widely
accepted. Can we continue to create a symbol for each number we know? Where
should I stop creating new symbols and try to reuse – at 9, at 16, at 20 or 60?

What about other ways of writing the number - Tally marks or Roman numerals? Tally marks were the most primitive representations
- they resembled counting on the five fingers on a hand. Roman numerals are
representations with a combination of alphabets.

The Tally marks and Roman representations have a problem of
scalability and processing. It becomes extremely hard to write down large
numbers and perform arithmetic operations on them. As the system evolves, one
would be forced to move to an advanced representation system with placeholders and repeating symbols (like the units, tens, hundreds, etc... in our
commonly used decimal system).

The placeholders give importance or weight to the value digit.
The placeholder itself can be any value – 2, 8, 10, 16, 60, etc… For example,
if the placeholder is SIXTY, then the first symbol would represent the number
itself, the second symbol would represent SIXTY times the symbol value and so on. Though
this looked like an elegant system, the problem it posed is handling blanks!

For example, if I have to represent sixty in a placeholder system of sixty, then I would
have write the symbol for ‘one’ and then leave a blank to denote the first placeholder.
Sixty = ‘1 ’. It would be difficult to differentiate between ‘1 ’ and ‘1’ – except
for the context and some guesswork.

The two questions that needed answers to complete the
numeral system were: How many symbols
are enough and what do we do with these non-existent values or blanks?

Providing answers to these two critical questions has been India’s biggest contribution to ‘mathematics’
and hence many advanced scientific inventions and discoveries.

From the Vedic times, a lot of scriptures were already
referring to large and small numbers with a placeholder value of 10.

One = eka;

Ten = dasha; Eleven = ekadasha (one plus ten)

Twenty = vimsatih; Twenty one = ekavimsatih (one plus
twenty)

Hundred = shat; two hundred = dwi-shat

And so on…

The second question needed an answer to ensure written
representation was non-ambiguous. This is where ‘shunya’ or zero came into
existence! But why was it such a big deal? We will look into this in the next
post…

What is this about?

It all started by a thought that crossed my mind in May 08. I completed 10 years of my job in Bangalore and it seemed that a technological generation gap had started emerging. Like the older generation who even today think that mobile phones are dispensable, I had not appreciated the power of e-communication!

As soon as you put your attention on some aspect of life, you start seeing only those things around you. There has been so much talk about celebrity blogs and also discovered that my teacher had taken this up. So here I am - presenting my thoughts and ideas. Happy reading and enjoy.